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  1. (, Nagoya Mathematical Journal)
    Abstract This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus$$4$$in every prime characteristic. More generally, the main result of the paper is that, for every$$g \geq 4$$and primep, every Newton polygon whosep-rank is at least$$g-4$$occurs for a smooth curve of genusgin characteristicp. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves. 
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    Free, publicly-accessible full text available March 1, 2026
  2. ; (, Communications in Algebra)
    Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q. 
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  3. ; ; ; ; ; ; ; (, Mathematics of Computation)
    Abstract. We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C ′ are curves over a finite field K, with K-rational base points P and P ′ , and let D and D ′ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C ′ , P ′ ) are doubly isogenous if Jac(C) and Jac(C ′ ) are isogenous over K and Jac(D) and Jac(D ′ ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than na¨ıve heuristics predict, and we provide an explanation for this phenomenon. 
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  4. ; ; (, Israel Journal of Mathematics)
    Abstract. Information about the absolute Galois group G K of a number field K is encoded in how it acts on the ´etale fundamental group π of a curve X defined over K. In the case that K = Q ( ζ n ) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of G K on the ´etale homology with coefficients in Z/nZ. The ´etale homology is the first quotient in the lower central series of the ´etale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of G K on all degrees of the associated graded quotient of the lower central series of the ´etale fundamental group of the Fermat curve of degree n, with coefficients in Z/nZ. 
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  5. ; ; (, Nagoya Mathematical Journal)
    Abstract. We study the p-rank stratification of the moduli space of cyclic degree ! covers of the projective line in characteristic p for distinct primes p and !. The main result is about the intersection of the p-rank 0 stratum with the boundary of the moduli space of curves. When ! = 3 and p ≡ 2 mod 3 is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type (r,s), and p-rank f satisfying the clear necessary conditions. 
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